Showing papers by "Herman Chernoff published in 1981"
TL;DR: The following inequality is useful in studying a variation of the classical isoperimetric problem as discussed by the authors : if the distribution of the variables is normal with mean 0 and variance 1, then the inequality holds if and only if the variables are linear in the distribution.
Abstract: The following inequality is useful in studying a variation of the classical isoperimetric problem. Let $X$ be normally distributed with mean 0 and variance 1. If $g$ is absolutely continuous and $g(X)$ has finite variance, then $E \{\lbrack g'(X)\rbrack^2\} \geq \operatorname{Var}\lbrack g(X)\rbrack$ with equality if and only if $g(X)$ is linear in $X$. The proof involves expanding $g(X)$ in Hermite polynomials.
250 citations
TL;DR: In this article, the authors considered the problem of comparing two treatments in a clinical trial and proposed a Bayes sequential procedure for a continuous time verion of the problem, where the heat equation plays a prominent role.
Abstract: SUMMARY A general approach to sequential decision-theoretic problems is illustrated with Anscombe's formulation of the problem of comparing two treatments in a medical trial. The Bayes sequential procedure for a continuous time verion of the problem is explicitly determined. Suboptimal procedures are proposed and evaluated; asymptotic results and numerical descriptions are provided. The continuous time version is found to provide accurate approximations even for clinical trials involving relatively small horizon sizes. A natural formulation for many statistical problems is one combining Bayesian, sequential and decision-theoretic aspects. For deciding the sign of a normal mean, Chernoff (1961, 1965a, 1965b), Breakwell & Chernoff (1964) and Bather (1962) replaced sums of successive observations by a continuous time Wiener process. The heat equation plays a prominent role in the resulting analysis, since optimal procedures may be characterized in terms of solutions of free boundary problems involving the heat equation. It would seem that the formidable appearance of some of the analysis involved in this approach has distracted potential users from taking advantage of other aspects which are easy or routine and which contribute clarity, asymptotic results with important theoretical implications, and numerical descriptions of sequential procedures and the resulting risks. Recent work by Lai, Levin, Robbins & Siegmund (1980) and by Begg & Mehta (1979) on a model for sequential medical trials proposed by Anscombe (1963) has prompted us to investigate the same problem using the above approach. In this paper we give the optimal procedure and show how it compares with suboptimal procedures. Secondly we use Anscombe's model to illustrate how such results may be obtained for similar problems. The key feature is the asymptotic resemblance of the problem to an optimal stopping problem for a Wiener process. Since many sequential decision-theoretic problems can be framed in terms of efficient scores which are asymptotically normal, the generality of the approach should be apparent. The formulation we refer to as Anscombe's model is the following. There is a horizon of N patients to be treated by one of two available treatments. In the initial experimental phase, n pairs of patients are treated sequentially, with different treatments randomly assigned to the patients in each pair. The differences, Xi, in the values of a continuous outcome variable for the ith pair are assumed to be independently and normally
38 citations